We prove, that for each r ∈ N, n ∈ N and s ∈ N there are a collection {yi} 2s i=1 of points y2s < y2s−1 < • • • < y1 < y2s + 2π =: y0 and a 2π -periodic function f ∈ C (∞) (R), such thatand for each trigonometric polynomial Tn of degree ≤ n (of order ≤ 2n + 1), satisfying(2)holds, where cr > 0 is a constant, depending only on r. Moreover, we prove, that for each r = 0, 1, 2 and any such collection {yi} 2s i=1 there is a 2π -periodic function f ∈ C (r) (R), such that (−1) i−1 f is convex on [yi, yi−1], 1 ≤ i ≤ 2s, and, for each sequence {Tn} ∞ n=0 of trigonometric polynomials Tn, satisfying (2), we have lim supω4(f (r) , 1/n) = +∞, where ω4 is the fourth modulus of continuity.

Bernstein inequality made it possible to obtain a constructive characterization of the approximation of periodic functions by trigonometric polynomials T_n of degree n. Instead, the corollary of this inequality for algebraic polynomials P_n of degree n, namely, the inequality $||? P_n'|| ? n ||P_n||$, where $? · ? := ? · ?_[?1,1]$ and $?(x) := \sqrt{1-x^2}$, does not solve the problem obtaining a constructive characterization of the approximation of continuous functions on a segment by algebraic polynomials. Markov inequality $||P_n'|| ? n^2 ||P_n||$ does not solve this problem as well. Moreover, even the corollary $||?_n P_n'|| ? 2n ||P_n||$, where $?_n(x) := \sqrt{1-x^2+1/n^2}$ of Bernstein and Markov inequalities is not enough. This problem, like a number of other theoretical and practical problems, is solved by Dzyadyk inequality $|| P_n' ?_n^{1-k} || ? c(s) n|| P_n ?_n^{-s} ||,$ valid for each s ? R. In contrast to the Bernstein and Markov inequalities, the exact constant in the Dzyadyk inequality is unknown for all s ? R, whereas the asymptotically exact constant for natural s is known: c(s) = 1 + s + s^2; and for n ? 2s, s ? N, even the exact constant is known. In our note, this result is extended to the case s ? n < 2s.

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